Question

A particle with charge \( q \) moving with velocity \( \vec{v} = v_0 \hat{i} \) enters a region with magnetic field \( \vec{B} = B_1 \hat{j} + B_2 \hat{k} \). The magnitude of force experienced by the particle is:

• A. \( q v_0 (B_1 + B_2) \)
• B. \( q \sqrt{v_0 (B_1 + B_2)} \)
• C. \( q v_0 \sqrt{B_1^2 + B_2^2} \)
• D. \( q \sqrt{v_0 (B_1^2 + B_2^2)} \)

Hint:

When dealing with a cross product, the magnitude of the resulting vector is found by taking the product of the magnitudes of the individual vectors and the sine of the angle between them. In this case, the cross product gives a result involving the square root of the sum of the squares of the magnetic field components.

Answer 1

The Correct Option is C

The magnetic force on a charged particle is defined by the Lorentz force law: \[\vec{F} = q (\vec{v} \times \vec{B})\]Here, \( \vec{v} \) represents the particle's velocity, \( \vec{B} \) is the magnetic field, and \( q \) denotes the particle's charge. When the velocity is \( \vec{v} = v_0 \hat{i} \) and the magnetic field is \( \vec{B} = B_1 \hat{j} + B_2 \hat{k} \), the magnitude of the resulting force is calculated as: \[|\vec{F}| = q v_0 \sqrt{B_1^2 + B_2^2}\]Consequently, the magnitude of the force on the particle is \( q v_0 \sqrt{B_1^2 + B_2^2} \).